The Alfvén Mach number compares a characteristic flow speed with the Alfvén speed $v_A=B/\sqrt{{\mu }_0\rho }$. In an axisymmetric magnetohydrodynamic wind, the relevant poloidal quantity is $M_{Ap}=|{\mathbf{u}}_p|\sqrt{{\mu }_0\rho }/|{\mathbf{B}}_p|$. With magnetohydrodynamic mass loading $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$, its square is ${\mu }_0{\kappa }^2/\rho$. The Alfvén surface is the locus $M_{Ap}=1$, and a smooth crossing requires the corresponding Alfvén-surface regularity condition for an axisymmetric wind. The total-speed and poloidal-speed definitions should be distinguished when there is substantial rotation or a strong toroidal magnetic field.

For a subset $S$ of the nonzero countable limit ordinals, let $B_{\alpha }=1+\mathbb{Q}$ when $\alpha \in S$ and $B_{\alpha }=\mathbb{Q}$ otherwise. The order sum is an aleph-one-like linear order whose nonempty proper initial segments are $\mathbb{Q}$ or $\mathbb{Q}+1$. At the limit cut before block $\delta$, the complementary final segment has a least point exactly when $\delta \in S$. Club agreement of countable filtrations makes this condition invariant modulo a nonstationary difference. Disjoint stationary subsets of omega-one consequently supply $2^{{\aleph }_1}$ nonisomorphic orders.

For a normal linear model with known error variance ${\sigma }^2$, the scaled Akaike criterion differs by a model-independent constant from

Starting from a fitted model, compare admissible single-term additions and deletions within specified lower and upper scopes, retaining hierarchical lower-order terms when an interaction requires them. Select a change reducing the Akaike information criterion and repeat until no admissible change improves it. For a Gaussian normal linear model with $N$ observations and $k$ fitted mean coefficients, the criterion used by R's linear-model stepwise search is $N\log (\mathrm{RSS}/N)+2k$, up to constants common to the models. The search is greedy and need not find the best model in the whole scope. Ordinary coefficient tests in the selected model do not account for this model selection. The software arguments are documented in the MASS documentation.

For a strongly convex rational cone $\sigma$, the group generated by $S={\sigma }^{\vee }\cap M$ is $M$. Inverting its finitely many monomial generators gives $k[M]$, so $\mathrm{Spec}k[M]$ is a dense open algebraic torus in $\mathrm{Spec}k[S]$. The map ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ extends torus multiplication to the entire variety.

// Target: geometry-and-topology.bigb

For $r>1$ and $\gcd (r,q)=1$, the quotient of ${\mathbb{A}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{q}y)$ is a cyclic quotient surface singularity. Its toric model uses the first-quadrant cone in ${\mathbb{Z}}^2+\mathbb{Z}(1,q)/r$. Conversely, every singular full-dimensional affine toric surface over an algebraically closed field of characteristic zero has this form. Smooth cones and cones with a torus factor require separate treatment.

// Target: geometry-and-topology.bigb

An arbitrary real affine equation has steady states exactly when $\mathrm{rank}A=\mathrm{rank}[A\mid -c]$. Its solution set is $s_{\ast }+\ker A$. For a finite-dimensional time-independent Lindblad equation, physical stationary states always exist by time averaging trajectories. If $A$ is invertible the steady state is unique. Conversely, a nonzero traceless Hermitian stationary direction can be decomposed into the difference of two equally weighted density operators; time averaging their trajectories yields two distinct stationary density operators. Thus singularity of the physical Bloch matrix also obstructs physical uniqueness.

For a two-level density operator, rates ${\gamma }_{12}$ from level one to two and ${\gamma }_{21}$ in the reverse direction give ${\dot{p}}_1=-{\gamma }_{12}{p}_1+{\gamma }_{21}{p}_2$. Their sum sets the longitudinal population relaxation rate, and their ratio fixes the stationary population imbalance. If both rates vanish, populations are conserved.

Decay of the off-diagonal coherence of a two-level density operator. Population relaxation contributes half its total rate; additional dephasing contributes ${\gamma }_{\varphi }\ge 0$ in the usual completely positive two-level model. Thus transverse relaxation should not automatically be identified with pure dephasing alone.

For the upper-triangular subgroup of ${\mathrm{GL}}_n(F)$, conjugation by $\mathrm{diag}({\varpi }^{-(n-1)},\dots ,1)$ expands the upper-unipotent coordinates. For a decomposed compact-open subgroup $K=T_0{U}_0$, its conjugates eventually contain $K$, while their fixed-vector spaces are isomorphic. Finite dimension therefore forces those spaces to be equal. Inverse conjugation contracts any fixed unipotent element into $U_0$, proving that the unipotent radical acts trivially in every admissible representation of this triangular group.

When cohorts share one time-since-entry survival distribution, their observations can be pooled using the Kaplan–Meier estimator, even when later cohorts have shorter administrative follow-up. A short-follow-up cohort still improves estimation of early survival, which is a factor in survival at later times. Pooling requires comparable survival across cohorts and independent censoring; calendar-period or cohort effects can invalidate that assumption.

For a fixed formal pseudodifferential operator $P={\sum }_j{p}_j{\partial }^j$ and a monic differential operator $L={\partial }^m+{\sum }_{l=0}^{m-1}{u}_l{\partial }^l$, cyclicity of the Adler trace gives $l_P(L)=\mathrm{Tr}(PL)$ with $c=\int p_{-m-1}dx$ and $a_l=\mathrm{res}({\partial }^{l}P)={\sum }_{k=0}^l\left(\genfrac{}{}{0ex}{}{l}{k}\right)p_{k-l-1}^{(k)}$. The function is affine in the variable coefficients; its variational derivative with respect to $u_l$ is $a_l$.

Replacing a shape-one-half gamma claim law by an exponential distribution with the same mean overestimates the adjustment coefficient. Indeed $R_I-R=\beta [3(1+\rho )-\sqrt{(\rho +9)(\rho +1)}]/[2(1+\rho )]>0$ because the difference of the squared terms is $8\rho (1+\rho )$. The resulting putative Lundberg inequality bound $e^{-R_{I}u}$ is smaller than the valid true-model bound $e^{-Ru}$ for $u>0$; the former is not justified for the true claim law. The true variance is twice the assumed variance, and its exponential tail decays more slowly.

Let $G$ be connected and compact with nonzero irreducible real Adjoint representation. Its Lie algebra has no proper invariant Lie algebra ideal. If it is Abelian, irreducibility forces dimension one, so $G$ is the circle group. Otherwise its centre has zero-dimensional Lie algebra and is finite. A proper closed normal subgroup likewise has zero-dimensional Lie algebra, hence is finite; connectedness of $G$ makes conjugation on that finite subgroup trivial, so the subgroup is central. A Lie group homomorphism onto a nontrivial connected target with surjective differential consequently has finite central group kernel. The qualifications “proper” and “nontrivial” are essential.

For a densely defined operator on a Hilbert space, $h$ belongs to $D(T^{\ast })$ exactly when the functional $f\mapsto ⟨Tf,h⟩$ is bounded in the ambient Hilbert norm on $D(T)$. Density makes its representing vector $T^{\ast }h$ unique. Differential expressions alone do not determine this domain.

// Target: analysis.bigb

Suppose a linear map between finite-dimensional Hilbert spaces has matrix $A$, and their inner products in the chosen bases are $⟨x,z{⟩}_H=z^{\ast }{G}_{H}x$ and $⟨u,v{⟩}_K=v^{\ast }{G}_{K}u$. The Gram matrices $G_H,G_K$ are positive-definite matrices. Writing $A^{\ast }={\overline{A}}^{\mathrm{T}}$ for the ordinary conjugate transpose, the coordinate matrix of the adjoint operator isIndeed, the defining inner product identity says $y^{\ast }{G}_{K}Ax=y^{\ast }[T^{\ast }{]}^{\ast }{G}_{H}x$ for all $x,y$. Thus simply transposing and conjugating is valid in orthonormal bases, where both Gram matrices are the identity matrix.

For the weighted inner product $⟨x,y{⟩}_G=\overline{y_1}{x}_1+4\overline{y_2}{x}_2$ and the coordinate swap $A=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, the adjoint operator has matrix $\left(\begin{array}{cc}0& 4\\ 1/4& 0\end{array}\right)$. With $x=(1,0{)}^{\mathrm{T}}$ and $y=(0,1{)}^{\mathrm{T}}$, both inner products equal $4$. The coordinate swap is its own ordinary transpose, but is not self-adjoint in this weighted inner product.

For Lie-algebra-valued fields and a constant invariant bilinear form on a Lie algebra, the connection terms in the displayed identity cancel. Indeed, invariance gives $\kappa ([A,Y],Z)+\kappa (Y,[A,Z])=0$. Ordinary integration by parts then yields $\int \kappa (D_{\mu }Y,Z)=-\int \kappa (Y,D_{\mu }Z)$ when boundary terms vanish, for example for compactly supported variations. This preserves covariance in variational calculations of Yang-Mills equations.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and matter transformation ${\varphi }^{\prime }=h\varphi$. Covariance $D_{\mu }^{\prime }{\varphi }^{\prime }=h{D}_{\mu }\varphi$ forces $A_{\mu }^{\prime }=h{A}_{\mu }{h}^{-1}+(i/g)({\partial }_{\mu }h)h^{-1}$. For $h=1-ig\omega +O({\omega }^2)$, expansion gives the displayed adjoint covariant derivative variation. With an orthonormal Hermitian Lie-algebra basis, $\delta A_{\mu }^a={\partial }_{\mu }{\omega }^a-g{f}^{abc}{A}_{\mu }^b{\omega }^c$. Thus Lorenz-gauge variation is $\delta (\partial \cdot A)=\partial \cdot \mathcal{D}\omega$. Choosing the Faddeev-Popov operator as $\mathcal{M}=-\partial \cdot \mathcal{D}$, with its field-independent sign absorbed in normalization, gives ghost action $-\bar{c}\partial \cdot \mathcal{D}c$; consistent propagator and vertex signs follow from this choice.

On the sl2 Lie algebra with basis $e=E_{12},h=E_{11}-E_{22},f=E_{21}$, the integral adjoint root polynomials areThese are conjugation by the upper and lower elementary matrices of the special linear group. Those matrices generate $S{L}_2(K)$, and a matrix commuting with both $E_{12}$ and $E_{21}$ is scalar. Thus the kernel is precisely $\{aI:a^2=1\}$ over every field, even in characteristic two, and the image is the displayed quotient.

When $\beta \rho (A)<1$, summing the adjacency-matrix bound for a discrete SIR epidemic gives the displayed Neumann series bound on the expected total infected count $Z$. For an undirected graph, the adjacency matrix of a graph is a symmetric matrix, so its Euclidean operator norm equals its spectral radius. The Cauchy-Schwarz inequality then yields $\mathbb{E}Z\le \sqrt{nm}/(1-\beta \rho )$ for $m$ initially infected vertices.